An index formula for the two variable Jordan block

Abstract

Abstract. On Hardy space H 2 (D 2 ) over the bidisk, let (S z , S w ) be the compression of the pair (T z , T w ) to the quotient module H 2 (D 2 ) M . In this paper, we obtain an index formula for (S z , S w ) when it is Fredholm. It is also shown that the evaluation operator L(0) is compact on a Beurling type quotient module if and only if the corresponding inner function is a finite Blaschke product in w.

A Brief Survey of Operator Theory in H 2(𝔻2)

“…There is a result without the assumption of Hilbert-Schmidtness. Theorem 6.5 (Lu, R. Yang and Y. Yang [67]). Let N = M ⊥ be a quotient module.…”

“…Lu, R. Yang and Y. Yang[67]). Let N = M ⊥ be a quotient module.If (S 1 , S 2 ) is Fredholm then both M ⊖ (z 1 M + z 2 M)and ker S 1 ∩ ker S 2 are finite dimensional and ind(S 1 , S 2 ) = dim (M ⊖ (z 1 M + z 2 M)) − dim (ker S 1 ∩ ker S 2 ) − 1.…”

Hilbert-Schmidtness of some finitely generated submodules in H2(D2)

“…For earlier work on the index of (S z 1 , S z 2 ) we refer readers to [20,21] and the references therein. Observe that…”